Problem: Factor the following expression: $x^2 - 2x - 3$
Explanation: When we factor a polynomial, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + a)(x + b) &=& xx &+& xb + ax &+& ab \\ \\ &=& x^2 &+& {(a + b)}x &+& {ab} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + a)(x + b) }&\hphantom{=}&\hphantom{ xx }&\hphantom{+}&\hphantom{ (a + b)x }&\hphantom{+}& \\ &=& x^2 & & {-2}x& & {-3} \end{eqnarray} $ The coefficient on the $x$ term is $-2$ and the constant term is $-3$ , so to reverse the steps above, we need to find two numbers that add up to $-2$ and multiply to $-3$ You can try out different factors of $-3$ to see if you can find two that satisfy both conditions. If you're stuck and can't think of any, you can also rewrite the conditions as a system of equations and try solving for $a$ and $b$ $ {a} + {b} = {-2}$ $ {a} \times {b} = {-3}$ The two numbers $-3$ and $1$ satisfy both conditions: $ {-3} + {1} = {-2} $ $ {-3} \times {1} = {-3} $ So we can factor the expression as: $(x {-3})(x + {1})$